Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ (―1)

Verified step by step guidance

Recall that the function \( \sin^{-1}(x) \), also known as arcsine, gives the angle \( y \) whose sine is \( x \), with the principal value range \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \).

Set up the equation \( y = \sin^{-1}(-1) \), which means \( \sin(y) = -1 \).

Identify the angle \( y \) within the principal range where the sine value is \( -1 \).

Recall the unit circle values: sine equals \( -1 \) at \( y = -\frac{\pi}{2} \) (or \( 270^\circ \) if considering degrees, but we use radians here).

Conclude that the exact value of \( y \) is \( -\frac{\pi}{2} \), since it lies within the principal range of arcsine.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted as sin⁻¹ or arcsin, returns the angle whose sine is a given number. It is defined for inputs between -1 and 1, and its output range is limited to angles between -π/2 and π/2 radians (or -90° to 90°).

Domain and Range of the Sine Function

The sine function outputs values only between -1 and 1, so the inverse sine function can only accept inputs within this range. Understanding the domain and range helps determine if a solution exists and what possible angle values correspond to a given sine value.

Exact Values of Sine at Special Angles

Certain angles have well-known sine values, such as sin(-π/2) = -1. Recognizing these special angles allows you to find exact values without a calculator, which is essential for solving inverse trigonometric equations precisely.

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